1. Quadratic Functions

2. A quadratic function is a function that can be written in the form of
Where a, b, and c are all real numbers and The domain of a quadratic function consists of all real numbers
This is a quadratic in standard form

3. a>0 a<0 Standard form of a parabola Axis of symmetry
vertex
vertex

4. Graph, find the vertex, axis of symmetry , intercepts and the domain and range
Find the y-intercept
Reflect over the axis of symmetry.

5. Graph, find the vertex, axis of symmetry , intercepts and the domain and range
Find the x-intercept.

6. Graph, find the vertex, axis of symmetry , intercepts and the domain and range
Find the x-intercept.

7. At x=2 there is a maximum value. The maximum value is 5. f(2)=5
Graph, find the vertex, axis of symmetry , intercepts and the domain and range
Axis of symmetry
Vertex
At x=2 there is a maximum value. The maximum value is 5. f(2)=5

8. Graph, find the vertex, axis of symmetry , intercepts and the domain and range
Find the y-intercept
Reflect over the axis of symmetry.

9. Graph, find the vertex, axis of symmetry , intercepts and the domain and range
Find the x-intercept

10. At x=-1 there is a minimum value. The minimum value is -9. f(-1)=-9
Graph, find the vertex, axis of symmetry , intercepts and the domain and range
Axis of symmetry
Vertex
At x=-1 there is a minimum value. The minimum value is -9. f(-1)=-9

11. A quadratic in general form
The equation for the axis of symmetry is

12. Find the minimum or maximum value and determine where it occurs
Find the minimum or maximum value and determine where it occurs. Then find the functions domain and range.
x=5/4 is the axis of symmetry and 5/4 is the x-coordinate of the vertex

13. Substitute 5/4 in the equation to find the y-coordinate of the vertex
Find the minimum or maximum value and determine where it occurs. Then find the functions domain and range.
Substitute 5/4 in the equation to find the y-coordinate of the vertex
Vertex
At x = 5/4 there is a maximum value of 1/8

14. At x=5/4 there is a maximum value of 1/8. f(5/4)=1/8
Find the minimum or maximum value and determine where it occurs. Then find the functions domain and range.
Vertex
At x=5/4 there is a maximum value of 1/8. f(5/4)=1/8

15. Find the minimum or maximum value and determine where it occurs
Find the minimum or maximum value and determine where it occurs. Then find the functions domain and range.
Axis of symmetry

16. Find the intercepts.
Plug in zero for x to find the y-intercept.
To find the x-intercept plug in zero for y.

17. Use the intercepts to graph.
Plug in zero for x to find the y-intercept.
To find the x-intercept plug in zero for y.

18. Graph using intercepts
To find the x-intercept plug in zero.
y-intercept.
Find the vertex

19. P , 9-15 odds, 17, 25, 31, 32, 39, 41

20. Y-int: plug in 0 for x X-int: plug in 0 for y
Graph using the intercepts and find the domain and range
Y-int: plug in 0 for x
X-int: plug in 0 for y

21. Graph using the intercepts and find the domain and range
Y-int
X-int:

22. If it won’t factor you need to use the quadratic equation.
Graph using the intercepts and find the domain and range
Y-int: plug in 0 for x
X-int: plug in 0 for y
If it won’t factor you need to use the quadratic equation.
If the answer is imaginary the graph doesn’t have x-intercepts.

23. Graph using the intercepts and find the domain and range
Y-int
X-int:
None

24. Day 2 Pg 313
10-16 even, 18, 26, 33, 37, 40, 42

25. How long does it take to for the ball to get to its maximum height?
The height s of a ball (in feet) thrown with an initial velocity of 100 feet per second from an initial height of 10 feet is given as a function of the time t (in seconds) by
How long does it take to for the ball to get to its maximum height?
What is the maximum height?

26. To find the maximum height, Find the vertex.
x represents t the time the ball is in the air.
y represents s the height of the ball.

27. The height s of a ball (in feet) thrown with an initial velocity of 100 feet per second from an initial height of 10 feet is given as a function of the time t (in seconds) by
How long does it take to for the ball to get to its maximum height?
3.125 seconds
What is the maximum height?
feet

28. How high is the ball after 2 seconds?
The height s of a ball (in feet) thrown with an initial velocity of 100 feet per second from an initial height of 10 feet is given as a function of the time t (in seconds) by
How high is the ball after 2 seconds?
After 2 seconds the ball is 146 ft high
How long before the ball hits the ground?
The ball hits the ground after 6.3 seconds

29. The maximum height of the ball is 36. 81 ft
The maximum height of the ball is ft. The maximum height occurs 59 ft from impact.
The defensive player must reach 8.72 ft to reach the punt

30. x = -1.67, 119.7
The ball hits the ground ft from the point of impact.
Graph the function that models the footballs parabolic path.

31. A farmer is going to fence in a rectangular area against a stream, he will not need to put fence where the stream is. He will also run fence in the middle of the rectangle perpendicular to the stream to divide it into two pens. He has purchased 1000 ft of fence, what dimensions should he make the pen to give his horses the most room? w
1000-3w
Since we want to find the pen with the most room the equation we need to maximize is the ____________________.
Area of the pen

32. A farmer is going to fence in a rectangular area against a stream, he will not need to put fence where the stream is. He will also run fence in the middle of the rectangle perpendicular to the stream to divide it into two pens. He has purchased 1000 ft of fence, what dimensions should he make the pen to give his horses the most room? w l
l= w
Since we want to find the pen with the most room the equation we need to maximize is the ____________________.
Area of the pen

33. A farmer is going to fence in a rectangular area against a stream, he will not need to put fence where the stream is. He will also run fence in the middle of the rectangle perpendicular to the stream to divide it into two pens. He has purchased 1000 ft of fence, what dimensions should he make the pen to give his horses the most room? w
l=1000-3w
The x coordinate of the vertex will give us the width at the maximum.

34. A farmer is going to fence in a rectangular area against a stream, he will not need to put fence where the stream is. He will also run fence in the middle of the rectangle perpendicular to the stream to divide it into two pens. He has purchased 1000 ft of fence, what dimensions should he make the pen to give his horses the most room?
w=166.7
l=1000-3w
Substitute the value for w into the side that is still unknown

35. A farmer is going to fence in a rectangular area against a stream, he will not need to put fence where the stream is. He will also run fence in the middle of the rectangle perpendicular to the stream to divide it into two pens. He has purchased 1000 ft of fence, what dimensions should he make the pen to give his horses the most room?
w=166.7
l=1000-3w
Substitute the value for w in for the side that is still unknown
The dimensions of the pen should be
166.7 ft x 500 ft

36. n=the first number m= the second number
Look for what your asked to find and assign variables to those values
n=the first number m= the second number
Write an equation for the value that is going to be minimized or maximized.
Look for information that you haven't used
Substitute into product equation
The two numbers whose product is a minimum and their difference is 10 are 5 and -5
The minimum product is -25

37. A rain gutter is made from sheets of aluminum that are 14 in
A rain gutter is made from sheets of aluminum that are 14 in. wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area? x
14 - 2x y x x x
Area = xy
14 in
y=14-2x y

38. A rain gutter is made from sheets of aluminum that are 14 in
A rain gutter is made from sheets of aluminum that are 14 in. wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area? x
14 - 2x x
3.5 in x x
Area = xy
14 in
y=14-2x
Area =
The depth is 3.5 in. and the maximum area is 24.5 sq. in.

39. P , 59 a,b, 61,65,71

40. P , 59 a,b, 61,65,71